STRAIGHT LINE:  Linear Equation or Linear Function

A straight line is one of the most basic curves in both geometry and algebra. In algebra, the basic equation of a straight line is:

y  =  mx  + b

where "m" is the slope of the straight line and "b" is the y-intercept of the straight line. Any equation that can be put into this (above) format is a straight line -and- all straight lines can (ultimately) be put into the format of this equation. This particular equation is known as the" "slope-intercept form" of a straight line.

The slope "m" of a straight line is a measure of the tilt or angle of the straight line when graphed in a rectangular coordinate system. Slope is defined as "rise over run." If any two points on the straight line are picked ... say (x1,y1) and (x2,y2) ... and the rise is computed by taking the y-value of the second point minus the y-value of the first point (y2 - y1) ... and this is divided by the run which is computed by taking the x-value of the second point minus the x-value of the first point (x2 - x1) ... the slope of the line will be found.

Slope = m  =  ( y2 - y1) / ( x2 - x1 ) = (change in y) / ( change in x )

The slope of a straight line is always constant ... no matter what two points are selected to compute it. A negative slope indicates that the line is moving downward to the right and decreasing; a zero slope indicates a line is perfectly horizontal and level; a positive slope indicates that a line is moving upward to the right and increasing; no slope (infinite slope is some textbooks) indicates that a line is perfectly vertical.

The Y-Intercept of a straight line is the intersection point of the straight line with the y-axis when graphed in a rectangular (Cartesian) coordinate system. It is technically that point which corresponds to an x-value of 0 (zero) ... the matched y-value in the equation when "x" is set equal to zero.

Example:  Given the equation 2x + 3y = 9, describe the equation.

Procedure ... solve the equation for the variable "y" ... subtract "2x" from both sides, divide both sides by "3" ... yielding:

y  =  -2x / 3  +  3   -or-   y  =  (-2/3)x  +  3

This equation is clearly of the form "y = mx + b" and, therefore, must be a straight line. The slope is the x-coefficient which is -2/3; the y-intercept is the constant term 3.

Description: The equation is a straight line. The slope of the straight line is -2/3. The y-intercept of the straight line is 3. The line is moving downward to the right (decreasing) when graphed in a coordinate system. The domain of the curve (the x values which make sense for the equation) is all real numbers. The domain of the curve (the y-values produced by substituting in all allowed x-values) is all real numbers.

To graph the curve, you would make an x-y coordinate system and then locate the y-intercept first. From the y-intercept you would mark off the slope ... going down 2 and over to the right 3 ... locating a second point of the line. You would join the y-intercept and this second point with a straight line.

Other Info: Two straight lines are parallel if and only if they have the same slopes. Two straight line are perpendicular if and only if they have opposite reciprocal slopes (i.e: if the slope of the first is -2/3 then the other is perpendicular if it has a slope of +3/2).


 


Return to Index page